# 置换的奇偶性

$\sgn(\sigma)=(-1)^{N(\sigma)}$

$\sgn(\sigma)=(-1)^m$

## 例子

$\sigma=(2 3) (1 2) (2 4) (3 5) (4 5),\;$

## 性质

• 两个偶置换的复合是偶的
• 两个奇置换的复合是偶的
• 一个奇置换与偶置换的复合是奇的

• 任何偶置换的逆是偶的
• 任何奇置换的逆是奇的

$\operatorname{sgn} : S_n \to \{-1,1\}$

(a b c d e) = (a e) (b e) (c e) (d e)

## 两个定义的等价性

### 证明一

σ = T'1 T'2 ... T'k'
σ = Q'1 Q'2 ... Q'm'

(2 5) = (2 3)(3 4)(4 5)(4 3)(3 2)

σ = T1 T2 ... Tk
σ = Q1 Q2 ... Qm

### 证明二

$P(x_1,\ldots,x_n)=\prod_{i

$P(x_1, x_2, x_3) = (x_1 - x_2)(x_2 - x_3)(x_1 - x_3).\;$

$\operatorname{sgn}(\sigma)=\frac{P(x_{\sigma(1)},\ldots,x_{\sigma(n)})}{P(x_1,\ldots,x_n)}$

$\operatorname{sgn}(\sigma\tau) = \frac{P(x_{\sigma(\tau(1))},\ldots,x_{\sigma(\tau(n))})}{P(x_1,\ldots,x_n)}$
$= \frac{P(x_{\sigma(1)},\ldots,x_{\sigma(n)})}{P(x_1,\ldots,x_n)} \cdot \frac{P(x_{\sigma(\tau(1))},\ldots, x_{\sigma(\tau(n))})}{P(x_{\sigma(1)},\ldots,x_{\sigma(n)})}$
$= \operatorname{sgn}(\sigma)\cdot\operatorname{sgn}(\tau)$

### 证明三

• $\tau_i^2 = 1$ 对所有i
• $\tau_i\tau_{i+1}\tau_i = \tau_{i+1}\tau_i\tau_{i+1}$  对所有i < n − 1，
• $\tau_i\tau_j = \tau_j\tau_i$  如果 |i − j| ≥ 2。